{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p26: Eigenvalues of 2nd order Chebyshev diff. matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='svg'\n",
    "from chebPy import *\n",
    "from numpy import dot,argsort,diag,real,imag,pi,array,polyfit,polyval,zeros\n",
    "from numpy.linalg import eig\n",
    "from matplotlib.pyplot import figure,loglog,semilogy,plot,title,ylabel,text,xlim"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/praveen/Applications/anaconda/lib/python2.7/site-packages/numpy/lib/polynomial.py:595: RankWarning: Polyfit may be poorly conditioned\n",
      "  warnings.warn(msg, RankWarning)\n"
     ]
    },
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   "source": [
    "N = 60; D, x = cheb(N); D2 = dot(D,D); D2 = D2[1:N,1:N]\n",
    "Lam, V = eig(D2)\n",
    "ii = argsort(-Lam); e = Lam[ii]; V = V[:,ii]\n",
    "\n",
    "# Plot eigenvalues\n",
    "figure(figsize=(8,4))\n",
    "loglog(-e,'o')\n",
    "semilogy(2*N/pi*array([1,1]),array([1,1e6]),'--')\n",
    "ylabel('eigenvalue')\n",
    "title('N='+str(N)+'      max |$\\lambda$| = '+str(max(-e)/N**4)+'$N^4$')\n",
    "text(2.1*N/pi,24,'$2\\pi/N$')\n",
    "\n",
    "# Plot eigenmode N/4 (physical)\n",
    "figure(figsize=(8,4))\n",
    "vN4 = zeros(N+1)\n",
    "vN4[1:N] = V[:,N/4];\n",
    "xx = arange(-1.0,1.01,0.01)\n",
    "vv = polyval(polyfit(x,vN4,N),xx)\n",
    "plot(xx,vv,'-')\n",
    "plot(x,vN4,'o')\n",
    "xlim((-1.0,1.0))\n",
    "title('eigenmode N/4')\n",
    "\n",
    "# Plot eigenmode N (nonphysical)\n",
    "figure(figsize=(8,4))\n",
    "vN = V[:,N-2]\n",
    "semilogy(x[1:N],abs(vN))\n",
    "plot(x[1:N],abs(vN),'.')\n",
    "xlim((-1.0,1.0))\n",
    "title('absolute value of eigenmode N (log scale)');"
   ]
  }
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